I have to do a convolution of a periodic signal with a Dirac impulse.
$\quad \quad x(t)=\sin(π\, t)(u(t)−u(t−2))$
$\quad \quad h(t)=u(t−1)−u(t−3)$
The first is a periodic signal that intersects the x-axis at points 0, 1, 2, .... The second is a rectangle (Dirac impulse) that intersects the x-axis at points 1 and 3.
For $t−1 < 0$ and $t−3 > 0$ the convolution doesn't exist.
I think the convolution is $\int_0^1\sin \pi\, (t-\tau)\, dt - \int_1^{t-1}\sin \pi\, (t-\tau)\, dt$, but my book gives $\int_0^1\sin \pi\, t\, dt$
Could someone tell me why he cancel -τ, and how to do this example without using a Fourier transform? Sorry for my terrible english, but i'm italian.